The method of fundamental solutions for analytic functions in complex analysis

Xiaoguang Yuan; Quan Jiang; Zhidong Zhou; Fengpeng Yang; Xiaoguang Yuan; Quan Jiang; Zhidong Zhou; Fengpeng Yang

doi:10.3934/math.2022380

AIMS Mathematics

2022, Volume 7, Issue 4: 6820-6851. doi: 10.3934/math.2022380

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The method of fundamental solutions for analytic functions in complex analysis

1.
School of Transportation and Civil Engineering, Nantong University, Nantong, 226019, China
2.
School of Science, Nantong University, Nantong, 226019, China
3.
College of Materials, Xiamen University, Xiamen, 361005, China
4.
School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiaotong University, Shanghai, 200240, China

Received: 22 November 2021 Revised: 05 January 2022 Accepted: 09 January 2022 Published: 26 January 2022
MSC : 30E10, 35E05, 65E05, 65N80

This paper extends the method of fundamental solutions (MFS) for solving the boundary value problems of analytic functions based on Cauchy-Riemann equations and properties of harmonic functions. The conformal mapping technique is applied to introduce the singularities of the approximate analytic functions and reconstruct the fundamental solutions. The presented method can naturally introduce the information of homogeneous boundary conditions and singularity properties, when the conformal mapping technique or the reconstructed fundamental solutions are used. The numerical examples show that the proposed method has the advantages of conciseness, reliability, efficiency, high accuracy and easy-using, respectively. The developed method can be used to solve the boundary value problems (BVPs) of analytic functions without considering single-valuedness, which simplify the numerical analysis.
- method of fundamental solutions,
- analytic function,
- boundary value problem,
- conformal mapping
Citation: Xiaoguang Yuan, Quan Jiang, Zhidong Zhou, Fengpeng Yang. The method of fundamental solutions for analytic functions in complex analysis[J]. AIMS Mathematics, 2022, 7(4): 6820-6851. doi: 10.3934/math.2022380

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Abstract

This paper extends the method of fundamental solutions (MFS) for solving the boundary value problems of analytic functions based on Cauchy-Riemann equations and properties of harmonic functions. The conformal mapping technique is applied to introduce the singularities of the approximate analytic functions and reconstruct the fundamental solutions. The presented method can naturally introduce the information of homogeneous boundary conditions and singularity properties, when the conformal mapping technique or the reconstructed fundamental solutions are used. The numerical examples show that the proposed method has the advantages of conciseness, reliability, efficiency, high accuracy and easy-using, respectively. The developed method can be used to solve the boundary value problems (BVPs) of analytic functions without considering single-valuedness, which simplify the numerical analysis.

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